Optimal Experimental Design and Estimation when Potential Outcomes are Bounded

2026-08-10

AI Summary

This study addresses the joint optimization of experimental design and estimation to minimize worst-case mean squared error (MSE) in finite populations with bounded potential outcomes, such as binary outcomes. By analyzing all assignment mechanisms within the class of affine estimators, the authors demonstrate that independent randomization coupled with an intercept-free regression using midpoint-centered covariates achieves optimal performance. This approach reduces the worst-case MSE by 50% compared to classical paired randomization with fixed-effects regression. Moreover, the optimality of this method is shown to extend beyond the affine class, establishing the theoretical superiority of independent random assignment in minimizing worst-case estimation error under bounded potential outcomes.

Abstract

I study the optimal design and analysis of randomized experiments for estimating finite-population average treatment effects when potential outcomes are known to be bounded, as with binary outcomes. Among all assignment mechanisms and a broad class of affine estimators, worst-case mean-squared error (MSE) is minimized by independent random assignment and an unconventional regression of the support-midpoint-centered outcome on the recentered treatment, with no intercept. This contrasts with the usual prescription of balanced complete randomization and difference-in-means estimation: when outcomes are bounded, randomness in the realized treatment share is informative. The worst-case gain over full-sample complete randomization is asymptotically small, but gains can be first-order relative to other designs: complete within-pair randomization and pair-fixed-effect regression have twice the worst-case MSE. I extend the result to allow for arbitrary estimators. Independent random assignment remains optimal, and the generally-nonlinear optimal estimator can meaningfully reduce worst-case MSE.

Problem

• Optimal Experimental Design
• Bounded Potential Outcomes
• Average Treatment Effect
• Worst-case MSE
• Randomized Experiments

Innovation

• optimal experimental design
• bounded potential outcomes
• worst-case MSE
• independent random assignment
• affine estimators